What is (\partial)/(\partial x)((z)(x^3z^5+sin(y^7z^5)+6)) ?

The answer to (\partial)/(\partial x)((z)(x^3z^5+sin(y^7z^5)+6)) is 3z^6x^2

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(\partial)/(\partial x)((z)(x^3z^5+sin(y^7z^5)+6))

\frac{\partial}{\partial x} ((z) (x^{3} z^{5} + sin (y^{7} z^{5}) + 6))

3 z^{6} x^{2}

Solution steps

\frac{\partial}{\partial x} ((z) (x^{3} z^{5} + sin (y^{7} z^{5}) + 6))

Treat

z, y

as constants

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= z \frac{\partial}{\partial x} (x^{3} z^{5} + sin (y^{7} z^{5}) + 6)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= z (\frac{\partial}{\partial x} (x^{3} z^{5}) + \frac{\partial}{\partial x} (sin (y^{7} z^{5})) + \frac{\partial}{\partial x} (6))

\frac{\partial}{\partial x} (x^{3} z^{5}) = 3 z^{5} x^{2}

\frac{\partial}{\partial x} (sin (y^{7} z^{5})) = 0

\frac{\partial}{\partial x} (6) = 0

= z (3 z^{5} x^{2} + 0 + 0)

Simplify

z (3 z^{5} x^{2} + 0 + 0) : 3 z^{6} x^{2}

= 3 z^{6} x^{2}

3 (3 d - 4 e) + 2 (d - 5 e)

\frac{1}{x \cdot ( x ^{2} + 1 )}

3 sin (\frac{π}{12} (t - 6)) + 13

\frac{2 x ^{3} + x ^{2} + 5 x - 2}{( x ^{2} + 1 ) ( x ^{2} + 4 )}

\frac{1}{4 ( x + 7 )}

What is (\partial)/(\partial x)((z)(x^3z^5+sin(y^7z^5)+6)) ?
The answer to (\partial)/(\partial x)((z)(x^3z^5+sin(y^7z^5)+6)) is 3z^6x^2